Estimation of Threshold Cointegration · 2015. 3. 2. · Sketch of Proof 1 Show √ n “ βˆ∗...

Estimation of Threshold Cointegration

Myung Hwan Seo

London School of Economics

December 2006

Seo Threshold Cointegration

http://www.lse.ac.uk

Model Asymptotics Inference Conclusion

Outline

1 ModelEstimation MethodsLiterature

2 AsymptoticsConsistencyConvergence RatesAsymptotic Distributions

3 Inference

4 Conclusion



Model Asymptotics Inference Conclusion Estimation Methods Literature

Outline



3 Inference

4 Conclusion




Model I

Variables:

xt : p-dimensional I (1) vector that is cointegrated with the cointegratingvector β. The first element of β is normalized to 1.zt (β) = x′tβ : equilibrium error, or error-correction termXt−1 (β) = (1, zt−1 (β) , ∆x′t−1, · · · , ∆x′t−l+1)

′: the regressor which is a

(pl + 2)-dimensional vector.

Two-regime threshold vector error correction model (Balke and Fomby1997)

∆xt =

A′Xt−1 (β) + ut,

(A + D)′ Xt−1 (β) + ut,if zt−1 (β) ≤ γif zt−1 (β) > γ

t = l + 1, ..., n. That is,

∆xt = A′Xt−1 (β) + D′Xt−1 (β) · 1 zt (β) > γ+ ut,

where 1 · is the indicator function.




Model II

Matrix notation: Let α = vec (A) and δ = vec (D) , where vec stacksrows of a matrix and

y =

∆xl+1...

∆xn

, u =

ul+1...

un

,

X (β) =

X′l (β)...

X′n−1 (β)

, X∗γ (β) =

X′l (β) · 1 zl (β) > γ...

X′n−1 (β) · 1 zn−1 (β) > γ

.

Then,u∗ (θ, β) = y− (X (β)⊗ Ip) α +

(X∗γ (β)⊗ Ip

)δ.




Estimation I

Classification of Parameters:

Long-run parameter: the cointegrating vector β

Short-run parameters: collectively we denote them as θ.

Slope parameters: A and D or α and δ

Threshold parameters: γ (β ?)

Estimation method:1 Least squares: denoted with the superscript *2 Smoothed Least squares (Seo & Linton 2005)




Estimation II

Least Squares Estimation

Objective Function:

S∗n (θ, β) = u∗ (θ, β)′ u∗ (θ, β) ,

where θ indicates all the short-run parameters (α′, δ′, γ)′.

LS estimator: “θ∗, β∗

”= arg min

θ,β

S∗n (θ, β) ,

where the minimum is taken over a compact parameter space.Concentrated LS estimator: for a fixed (β, γ) ,»

α∗ (β, γ)

δ∗ (β, γ)

–=

»X (β)′ X (β) X (β)′ X∗γ (β)X∗γ (β)′ X (β) X∗γ (β)′ X∗γ (β)

–−1„ X (β)′

X∗γ (β)′

«⊗ Ip

!y,

which is then plugged back into S∗n for optimization over (β, γ) .




Smoothed LS I

Smoothed Least Squares Estimation (Seo & Linton (2005))Define a bounded function K (·) satisfying that

lims→−∞

K (s) = 0, lims→+∞

K (s) = 1.

Let Kt (β, γ) = K“

zt(β)−γσn

”for σn → 0 and replace 1 zt (β) > γ with

Kt (β, γ) :

Xγ (β) =

0B@ X′l (β)Kl (β, γ)...

X′n−1 (β)Kn−1 (β, γ)

1CAu (θ, β) = y− (X (β)⊗ Ip) α + (Xγ (β)⊗ Ip) δ.

Then, the Smoothed Least Squares (SLS) estimator is“θ, β”

= arg minθ,β

Sn (θ, β) ,

where Sn (θ, β) = u (θ, β)′ u (θ, β) .




Smoothed LS II

Concentration»α (β, γ)

δ (β, γ)

–=

»X (β)′ X (β) X (β)′ Xγ (β)Xγ (β)′ X (β) Xγ (β)′ Xγ (β)

–−1„ X (β)′

Xγ (β)′

«⊗ Ip

!y.




Literature I

Applications of Threshold Cointegration:

(PPP) Baum, Barkoulas, & Caglayan (2001) , Taylor (2001) , Enders &Falk (1998) , O’Connell (1998) , Obstfeld & Taylor (1997) , Michael,Mobay, & Peel (1997)

(LOP) Lo & Zivot (2001) , O’Connell & Wei (1997) , Parsley & Wei(1996)

(Term structure) Balke & Wohar (1998) , Baum & Karasulu (1998) ,Martens, Kofman, & Vorst (1998) Enders & Siklos (2001)

(Long-run money demand) Escribano (2004)




Literature II

Stability of threshold cointegration model - Bec & Rahbek (2004) ,Saikkonen (2005)

Testing for threshold cointegration:

Cointegration : Enders & Siklos (2001) , Bec Guay & Guerre (2004), M.Seo (2005, 2006) , Park & Shintani (2005)Threshold effect (assuming the presence of cointegration) : Hansen & B.Seo (2002)

Estimation - No Distribution theory yet.Bec & Rahbek (2004)− known cointegrating vector.stationary threshold models:

discontinuous TAR: Chan (1993) , continuous TAR: Chan & Tsay (1998) ,subsampling: Gonzalo & Wolf (2005)Regression: Hansen (2000) , Seo and Linton (2005)

de Jong (2001, 2002) : smooth nonlinear error correction model



Model Asymptotics Inference Conclusion Consistency Convergence Rates Asymptotic Distributions

Outline



3 Inference

4 Conclusion




Review of Asymptotics for stationary threshold models I

Least Squares Smoothed LSModel Slope Threshold ThresholdDiscontinuous

√n rate n rate

pnσ−1

n rate(Chan) Normal complex Poisson NormalContinuous

√n rate

√n rate

√n rate

(Chan & Tsay) Normal Normal NormalDiminishing

√n rate n1−2α rate

pn1−2ασ−1

n rate(Hansen) Normal Function of BM Normal

If the cointegrating vector is known, these results will apply.




Consistency

Assumption (1)

(a) ut is an independent and identically distributed sequence withEut = 0, Eutu′t = Σ that is positive definite. Furthermore, it has a densitywrt Lebesque measure that is everywhere positive.(b) ∆xt, zt is a sequence of strictly stationary strong mixing randomvariables with mixing numbers αm, m = 1, 2, . . . , that satisfyαm = o

(m−(α0+1)/(α0−1)

)as m →∞ for some α0 ≥ 1, and for some ε > 0,

E∣∣XtX>t

∣∣α0+ε< ∞ and E |Xt−1ut|α0+ε

< ∞. Furthermore, E∆xt = 0 andx[ns]/

√n converges weakly to a vector Brownian motion B with the

covariance matrix , which is the long-run covariance matrix of ∆xt and hasrank p− 1 s.t. β′0 = 0.(c) E

[X′t−1D0D′0Xt−1|zt−1

]> 0 a.s.

Condition (c) implies the discontinuity of the model




Consistency

Theorem (1)

Under Assumption 1,√

n(β∗ − β0

)and

(θ∗ − θ0

)are op (1) . In addition,

assume that s2 |1 s > 0 − K (s)| < M < ∞, for any s ∈ R. Then,√

n(β − β0

)and

(θ − θ0

)are op (1) .

Sketch of Proof1 Show

√n“β∗ − β0

”= Op (1) by contradiction.

2 Show√

n“β∗ − β0

”and

“θ∗ − θ0

”are op (1) based on ULLN.

3 Show the difference between Sn and S∗n are asymptotically uniformlynegligible.




Convergence Rate

Long-run vs. Short-run parametersThreshold vs. Slope parametersSmoothed vs. Unsmoothed estimator

Theorem (2)

Under Assumption 1, β∗ = β0 + Op(n−3/2

)and γ∗ = γ0 + Op

(n−1).

In consequence, α∗ and δ∗ converge at the rate of√

n.




Asymptotic Distribution I

Assumption (2)

(a) E[|X′t ut|r] < ∞, E[|X′t Xt|r] < ∞, for some r > 4,(b) ∆xt, zt is a sequence of strictly stationary strong mixing randomvariables with mixing numbers αm, m = 1, 2, . . . , that satisfyαm ≤ Cm−(2r−2)/(r−2)−η for positive C and η,as m →∞.(c) For some integer h ≥ 1 and each integer i such that 1 ≤ i ≤ h, all z in aneighborhood of γ, almost every ∆, and some M < ∞, f (i) (z|∆) exists andis a continuous function of z satisfying

∣∣f (i) (z|∆)∣∣ < M. In addition,

f (z|∆) < M for all z and almost every ∆. (d) and the conditional jointdensity f (zt, zt−m|∆t,∆t−m) < M, for all (zt, zt−m) and almost all(∆t,∆t−m) .(e) θ0 is an interior point of Θ.




Asymptotic Distribution II

Assumption (3)

(a) K is twice differentiable everywhere,∣∣K(1) (·)

∣∣ and∣∣K(2) (·)

∣∣ areuniformly bounded, and each of the following integrals is finite:∫ ∣∣K(1)

∣∣4 ,∫ ∣∣K(2)

∣∣2 ,∫ ∣∣v2K(2) (v) dv

∣∣ .(b) For some integer h ≥ 1 and each integer i (1 ≤ i ≤ h) ,∫ ∣∣viK(1) (v) dv

∣∣ < ∞, and∫si−1sgn (s)K(1) (s) ds = 0,

and∫

shsgn (s)K(1) (s) ds 6= 0,

and K (x)−K (0) ≷ 0 if x ≷ 0.




Asymptotic Distribution III

Assumption (3 cont’d)

(c) For each integer i (0 ≤ i ≤ h) , and η > 0, and any sequence σnconverging to 0,

limn→∞

σi−hn

∫|σns|>η

∣∣∣siK(1) (s)∣∣∣ ds = 0,

and limn→∞

σ−1n

∫|σns|>η

∣∣∣K(2) (s)∣∣∣ ds = 0.

(d) lim supn→∞

nσ2hn < ∞ and

limn→∞

σ−2hn

∫|σns|>η

∣∣∣K(1) (s)∣∣∣ ds = 0.




Asymptotic Distribution IV

Assumption (3 cont’d)

(e) For some µ ∈ (0, 1], a positive constant C, and all x, y ∈ R,∣∣∣K(2) (x)−K(2) (y)∣∣∣ ≤ C |x− y|µ .

(f )For some sequence mn ≥ 1, and ε > 0,

log (nmn)(

n1−6/rσ2nm−2

n

)−1→ 0

σ−3k−1n n3/r+εαmn → 0.




Asymptotic Distribution V

Condition (f ) serves to determine the rate for σn. When the data arei.i.d. and the regressors possess a moment generating function, theconditions can be weakened to

log (n)

nσ2n

→ 0,

since αmn = 0 and we can set mn = 1 in this case. Although condition(e) provides permissible rates for the bandwidth selection, it may not besharp.




Asymptotic Distribution VI

Let B be the limit Brownian motion of the partial sum process of ∆xt, whosecovariance matrix is Ω, and

σ2v= E

[∥∥∥K(1)∥∥∥2

2

(X′t−1D0ut

)2+∥∥∥K(1)

∥∥∥2

2

(X′t−1D0D′0Xt−1

)2 |zt−1 = γ0

]f (γ0)

σ2q= K(1) (0) E

(X′t−1D0D′0Xt−1|zt−1 = γ0

)f (γ0) ,

where K(i) is the i-th derivative of K,K(1) (s) = K(1) (s) (1 s > 0 − K (s)) , and ‖g‖2

2 =(∫

g2)




Asymptotic Distribution VII

TheoremSuppose Assumption 1 - 3 hold. Let W denote a standard Brownian motionthat is independent of B.Then,(

nσ−1/2n

(β − β0

)√nσ−1

n (γ − γ0)

)d→ σv

σ2q

( ∫ 10 BB′

∫ 10 B∫ 1

0 B′ 1

)−1( ∫BdW

W (1)

),

√n(

α− α0

δ − δ0

)d→ N

(0,

[E(

1 dt−1dt−1 dt−1

)⊗ Xt−1X′t−1

]−1

⊗ Σ

)and they are asymptotically independent. The unsmoothed estimator α∗ andδ∗ have the same asymptotic distribution as α and δ.




Two-step Estimation I

Corollary

Let γ (β) be the smoothed estimator of γ when β is given. Then, γ(β∗)

has

the same asymptotic distribution as that of γ (β0) , which is N(

0,σ2

vσ4

q

).

Thus, we do not need to estimate the long-run variance for the inferencefor γ.

Suppose that βn = β + Op(n−1). For example, βn can be obtained

from the simple OLS of the first element of xt to the other elements ofxt, as in Engle-Granger procedure. Note that, by multipying β0 bothsides of the model

∆xt =

A′Xt−1 (β) + ut, if zt−1 (β) ≤ γ

(A + D)′ Xt−1 (β) + ut, if zt−1 (β) > γ,




Two-step Estimation II

we obtain a threshold autoregressive process for zt = x′tβ0

∆zt =

a0zt−1 + a1∆zt−1 + · · ·+ ut, if zt−1 ≤ γ

(a0 + d0) zt−1 + (a1 + d1) ∆zt−1 + · · ·+ ut, if zt−1 > γ

But, the plug-in estimate α (βn) and δ (βn) do not have the sameasymptotic distribution as α (β0) and δ (β0) .

To treat the estimate βn as the true value β0, we first need

1√n

∑t

∆xt−kKt−1 (βn, γn) x′t−1 (βn − β0) = op (1)




Two-step Estimation III

In case of Ezt = 0, we can still retain the Normality by replacing thezt−1

(β)

with

zt−1

(β)

= x′t−1β =

(xt−1 −

1n

∑s

xs−1

)′β,

for any n-consistent β, as in de Jong (2001) . It is worth noting,however, that the asymptotic variance increases by doing this.




Outline



3 Inference

4 Conclusion




Inference I

Asymptotic Variance Estimation:

long-run covariance matrix Ω : see Andrews (1991) for example.variance of threshold estimate (σv and σq) : Let

τt =1√σn

Xt−1

“β”′

DK(1)t−1

“β, γ

”ut, (1)

where ut is the regression residual, and let

σ2v =

1n

Xt

τ 2t , and σ2

q =σn

nQn22

“θ”

,

where Qn22 is the diagonal element corresponding to γ of the secondderivative of Sn. Refer to Seo and Linton (2005) .variance of slope estimate: the same as linear regression.

We do not have to estimate the density and conditional expectation by anonparametric method. Hansen (2000) uses a nonparametric method toestimate the asymptotic variance; the introduction of a smoothingparameter is necessary.




Inference II

Testing for two thresholds: To test the null of one threshold against twothreshold, we can extend the SupLM test statistic of Hansen and Seo(2002) that examines the null of no threshold. Due to the fastconvergence rate of the least squares estimator of β and γ, we can treatthe estimated β and γ as if known. For the p-value, the fixed regressorbootstrap and residual bootstrap can be employed as described there.




Outline



3 Inference

4 Conclusion




Conclusion

The paper establishes the consistency and convergence rates of the LSEand SLSE of the threshold vector error correction model. In particular,

The LSE of the cointegrating vector estimate converges extremely fast.This validates a two-step estimation, in which the cointegrating vector isfirst estimated by LS and the other short-run parameters by SLS.The LSE of the slope is asymptotically not affected by the estimation ofthe other parameters

The limit distribution of the SLSE is derived and thus providing a wayof inference for the cointegrating vector.

Multiple-Regime Threshold: We can estimate the thresholdssimultaneously or sequentially. Hansen (1999) and Bai and Perron(1998) .

More than one cointegrating relation



Estimation of Threshold Cointegration · 2015. 3. 2. · Sketch of Proof 1 Show √ n “ βˆ∗...

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